Non-autonomous vector integral equations with discontinuous right-hand side
Cubiotti, Paolo
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001), p. 319-329 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

We deal with the integral equation $u(t)=f(t,\int_I g(t,z)u(z)\,dz)$, with $t\in I:=[0,1]$, $f:I\times \Bbb R^n \to \Bbb R^n$ and $g:I\times I\to[0,+\infty[$. We prove an existence theorem for solutions $u\in L^s(I,\Bbb R^n)$, $s\in \,]1,+\infty]$, where $f$ is not assumed to be continuous in the second variable. Our result extends a result recently obtained for the special case where $f$ does not depend explicitly on the first variable $t\in I$.

Publié le : 2001-01-01
Classification:  45G10,  45P05,  47H15,  47J05,  47N20 
@article{119246,
     author = {Paolo Cubiotti},
     title = {Non-autonomous vector integral equations with discontinuous right-hand side},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {42},
     year = {2001},
     pages = {319-329},
     zbl = {1055.45004},
     mrnumber = {1832150},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119246}
}

Cubiotti, Paolo. Non-autonomous vector integral equations with discontinuous right-hand side. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 319-329. http://gdmltest.u-ga.fr/item/119246/

Aubin J.P.; Frankowska H. Set-Valued Analysis, Birkhäuser, Boston, 1990. | MR 1048347 | Zbl 1168.49014

Banas J.; Knap Z. Integrable solutions of a functional-integral equation, Rev. Mat. Univ. Complut. Madrid 2 (1989), 31-38. (1989) | MR 1012104 | Zbl 0679.45003

Cammaroto F.; Cubiotti P. Implicit integral equations with discontinuous right-hand side, Comment. Math. Univ. Carolinae 38 (1997), 241-246. (1997) | MR 1455490 | Zbl 0886.47031

Cammaroto F.; Cubiotti P. Vector integral equations with discontinuous right-hand side, Comment. Math. Univ. Carolinae 40 (1999), 483-490. (1999) | MR 1732487 | Zbl 1065.47505

Emmanuele G. About the existence of integrable solutions of a functional-integral equation, Rev. Mat. Univ. Complut. Madrid 4 (1991), 65-69. (1991) | MR 1142550 | Zbl 0746.45004

Emmanuele G. Integrable solutions of a functional-integral equation, J. Integral Equations Appl. 4 (1992), 89-94. (1992) | MR 1160090 | Zbl 0755.45005

Fečkan M. Nonnegative solutions of nonlinear integral equations, Comment. Math. Univ. Carolinae 36 (1995), 615-627. (1995) | MR 1378685

Hewitt E.; Stomberg K. Real and Abstract Analysis, Springer-Verlag, Berlin, 1965.

Himmelberg C.J.; Van Vleck F.S. Lipschitzian generalized differential equations, Rend. Sem. Mat. Univ. Padova 48 (1973), 159-169. (1973) | MR 0340692 | Zbl 0289.49009

Kantorovich L.V.; Akilov G.P. Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1964. | MR 0213845 | Zbl 0127.06104

Klein E.; Thompson A.C. Theory of Correspondences, John Wiley and Sons, New York, 1984. | MR 0752692 | Zbl 0556.28012

Lang S. Real and Functional Analysis, Springer-Verlag, New York, 1993. | MR 1216137 | Zbl 0831.46001

Naselli Ricceri O.; Ricceri B. An existence theorem for inclusions of the type $\Psi(u)(t)\in F(t,\Phi(u)(t))$ and application to a multivalued boundary value problem, Appl. Anal. 38 (1990), 259-270. (1990) | MR 1116184

Scorza Dragoni G. Un teorema sulle funzioni continue rispetto ad una e misurabili rispetto ad un'altra variabile, Rend. Sem. Mat. Univ. Padova 17 (1948), 102-106. (1948) | MR 0028385 | Zbl 0032.19702

Villani A. On Lusin's condition for the inverse function, Rend. Circ. Mat. Palermo 33 (1984), 331-335. (1984) | MR 0779937 | Zbl 0562.26002